The settings below are tested in production for 500 to 20.000 products and 3 mio attributes. To have a flexible and better server features management and also the best speed for to work with Prestasho,p take a dedicated or a VPS root server. Managed servers in most of the cases don't have the possibility to make own changes needed on the server configuration.

You need at least php 5.6.30. Better performance you will get with php 7 and mySQL 5.7 or mariaDB 10. There is no need for accelerators, in contrary, best is the use of php-fpm + opcache + ngnix proxied.

cageFS and another FIREWALL technique installed on the server configured correctly so that the back-office is not slow by blocking external services that Prestashop uses. If in doubt, or back-office slow, please ask the provider of your accommodation if there are firewalls installed !!!

PS 1.7.x in detail:

PS 1.7.0 > 1.7.3. - here you can still use php 5.6.30 (above and inclusive) without problems. For better performance use php 7.1. and mysql 5.7 or maria DB 10